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arXiv:2610.27416

Quantum algorithm for high-dimensional nonlinear stochastic differential equations based on Carleman linearization

Koichi Miyamoto

quant-phSubmitted Oct 20, 2026• Announced Oct 21, 2026• 0 citations
v1 · Oct 20, 2026v2 · Oct 30, 2026
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Abstract

Addressing nonlinear numerical problems in quantum computing remains challenging, but several approaches, including Carleman linearization (CL), have been proposed to construct quantum algorithms for such problems. In this work, we develop a CL-based method for a high-dimensional nonlinear stochastic differential equation (SDE) $\mathrm{d} \mathbf{X}_t = \left(A(t) \mathbf{X}_t + F(t) \mathbf{X}_t^{\otimes s}\right) \mathrm{d} t + B(t) \mathrm{d} \mathbf{W}_t$ for an $\mathbb{R}^N$-valued random process $\mathbf{X}_t$, aiming at quantum speed-up with respect to the dimension $N$. We introduce auxiliary processes $\mathbf{Y}^j_t$ that approximate $\mathbf{X}_t^{\otimes j}$, derive a linearized system of SDEs for them, and obtain the linear ordinary differential equation (ODE) governing their expectations. We then apply a quantum ODE solver to it and, from the resulting quantum state, extract an estimate for the expectation of a linear function of $\mathbf{X}_t$ by overlap estimation. We show that, provided the nonlinear and noise terms in the SDE are sufficiently small relative to the linear term, the CL error decreases exponentially with the truncation order $J$ in CL, and we can estimate the target expectation with a ${\rm polylog}(N)$ number of queries to unitaries encoding coefficient matrices $A$, $F$, and $B$. Finally, we also run a numerical demonstration for an SDE related to Langevin dynamics, observing the exponential decay of the CL error as $J$ increases.

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48 pages, 1 figure

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Nonlinear stochastic differential equations (NSDEs) are a pillar of mathematical modeling for scientific and engineering applications. Accurate and efficient simulation of large-scale NSDEs is prohibitive on classical computers due to the large number of degrees of freedom, and it is challenging on quantum computers due to the linear and unitary nature of quantum mechanics. We develop a quantum algorithm to tackle nonlinear differential equations driven by the Ornstein-Uhlenbeck (OU) stochastic process. The query complexity of our algorithm scales logarithmically with the error tolerance and nearly quadratically with the simulation time. Our algorithmic framework comprises probabilistic Carleman linearization (PCL) to tackle nonlinearity coupled with stochasticity, and stochastic linear combination of Hamiltonian simulations (SLCHS) to simulate stochastic non-unitary dynamics. We obtain probabilistic exponential convergence for the Carleman linearization of Liu et al. [1], provided the NSDE is stable and reaches a steady state. We extend deterministic LCHS to stochastic linear differential equations, retaining near-optimal parameter scaling from An et al. [2] except for the nearly quadratic time scaling. This is achieved by using Monte Carlo integration for time discretization of both the stochastic inhomogeneous term in LCHS and the truncated Dyson series for each Hamiltonian simulation.
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